Smith Chart Applications

The Smith chart is not merely a plotting tool but a powerful design instrument for impedance matching, stub design, and load characterization in RF and microwave circuits. Once the load impedance is located on the chart, the graphical operations replace complex algebraic calculations with simple rotations and intersections. This article explains how the Smith chart is applied to practical matching problems.

Core Concept Explanation

The primary application of the Smith chart is impedance matching. Any mismatch between a transmission line and its load means the load impedance point on the Smith chart is not at the center. The goal of matching is to move this point to the chart center (z = 1 + j0) by adding appropriate reactive elements or line sections. The Smith chart converts this design task into a graphical procedure.

The most common matching procedure involves two steps. First, find a position along the transmission line (rotating the load point clockwise toward the generator on the constant-|Γ| circle) where the real part of the normalized admittance equals 1, i.e., g = 1. At this point, only the susceptance component needs to be cancelled by a shunt stub. Alternatively, in impedance form, find the position where the normalized impedance has r = 1, and cancel the reactance using a series stub.

The Smith chart also handles admittance directly. When dealing with shunt (parallel) elements, it is convenient to work in normalized admittance y = Y/Y₀ = g + jb, where g is the normalized conductance and b is the normalized susceptance. The admittance Smith chart is identical in structure to the impedance chart but relabeled, with the g-circles and b-arcs replacing r-circles and x-arcs. The same physical chart can be used for admittance by rotating every impedance point by 180 degrees (λ/4 rotation).

Quarter-wave transformer design on the Smith chart is straightforward. Moving the load point exactly λ/4 (half revolution clockwise on the WTG scale) gives the impedance at the input of a λ/4 section. This graphically shows the transformation Z_in = Z₁²/Z_L.

Mathematical Expression

For shunt stub matching (single stub), the procedure uses admittance. After normalizing the load admittance y_L = Y_L/Y₀, rotate toward generator to find point where g = 1 (on the g = 1 circle in admittance form). Let the rotation angle correspond to distance d. At this point, the admittance is y = 1 + jb. The stub must provide susceptance -jb to cancel this residual, making the total admittance y_total = 1 + j0, which corresponds to the matched condition at the chart center.

The stub susceptance jb_stub is found by determining the length l of an open or short-circuited stub whose input admittance equals -jb. For a short-circuited stub: y_stub = -j·cot(βl). For an open-circuited stub: y_stub = j·tan(βl). These loci are traced along the outer edge of the Smith chart (where r = 0 or g = 0 circles are), and the length l is read from the WTG or WTL scale.

Practical Understanding

In microstrip circuit design, the Smith chart is used to design matching networks for low-noise amplifiers (LNAs), power amplifiers, and antenna feeds. The noise figure of an amplifier is minimized at a specific source impedance called the optimum noise impedance, which may differ from the conjugate match impedance. The Smith chart allows the designer to visually see both the noise circles and gain circles and choose a compromise matching point.

Another important application is determining the input impedance of a loaded transmission line at any point. Given the load, normalize it, plot it, rotate by the required electrical length, and read off the normalized impedance at the new point. Denormalizing gives the actual input impedance. This replaces the calculation of Z_in = Z₀ · [Z_L + jZ₀ tan(βl)] / [Z₀ + jZ_L tan(βl)].

Given:
Z₀ = 50Ω
Z_L = 25 − j50Ω
Find: stub position d and stub length l for shunt short-circuit stub matching

Why this formula applies:
Shunt stub matching requires working in admittance form to find g=1 circle intersection.

Formula:
y_L = Y_L/Y₀ = Z₀/Z_L
Rotate toward generator until g=1 on VSWR circle
Cancel residual susceptance with stub of admittance y_stub = −jb

Substitution:
y_L = 50/(25 − j50) = 50(25 + j50)/[(25)² + (50)²]
    = 50(25 + j50)/3125 = (1250 + j2500)/3125
    = 0.4 + j0.8

Plot y_L = 0.4 + j0.8 on Smith chart.
VSWR circle radius |Γ| = |z_L − 1|/|z_L + 1|
z_L = 1/y_L = 0.5 − j1.0
|Γ| = |(0.5 − j1 − 1)|/|(0.5 − j1 + 1)| = |−0.5 − j1|/|1.5 − j1|
    = √1.25/√3.25 = 1.118/1.803 = 0.620

Rotating from y_L along VSWR circle until g=1 circle:
g=1 intersection gives y = 1 + j1.58 (graphically)
Stub must provide b_stub = −1.58
For SC stub: b_stub = −cot(βl) = −1.58 → cot(βl) = 1.58 → βl = 0.567 rad
l = 0.567/(2π) × λ = 0.0902λ

Final Answer:
d ≈ 0.109λ from load toward generator
Short-circuit stub length l ≈ 0.09λ

Exam Tip: When using the Smith chart for stub matching, always convert to admittance (rotate 180° or use y-chart) before looking for the g=1 circle intersection for shunt stubs. The stub susceptance must exactly cancel the residual susceptance at that point. For series stub matching, stay in impedance form and find the r=1 circle intersection instead.

Mechanism Summary

• Convert load to normalized admittance y_L = Z₀/Z_L and plot on Smith chart. The VSWR circle is drawn with radius |Γ_L|.
• Rotate clockwise (toward generator) on the VSWR circle until intersecting the g = 1 circle. The rotation distance (read from WTG scale) gives the position d of the stub from the load.
• At the intersection, the admittance is 1 + jb. A shunt stub providing susceptance -jb is connected in parallel at position d, cancelling the residual susceptance.
• The stub length is found on the outer edge of the chart (r = 0 or g = 0 boundary) where the susceptance value -jb is located, and the WTG scale gives the stub length.
• Two intersection points with the g = 1 circle are generally found, giving two possible matching solutions with different d and l values. The shorter distances are preferred in practice.

Quick Revision

• Smith chart applications: impedance transformation, stub matching, input impedance calculation, VSWR and |Γ| determination.
• For shunt stub matching: work in admittance form (y = Y/Y₀), rotate to g=1 circle, cancel susceptance with stub.
• For series stub matching: work in impedance form (z = Z/Z₀), rotate to r=1 circle, cancel reactance with series stub.
• Admittance of a point = impedance of the point 180° opposite (λ/4 away) on the same VSWR circle.
• Stub lengths for SC stub: b = -cot(βl). For OC stub: b = tan(βl). These are read directly from the outer g=0 or r=0 arc of the chart.
• Common trap: using impedance form for shunt stubs instead of converting to admittance first, leading to incorrect stub position.
• Two solutions always exist for single stub matching. Both are valid; select based on minimum d or practical fabrication constraints.